Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#509 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
591 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(10,15)(11,19,21,38)(12,22)(13,24)(16,31,32,40)(17,29)(18,25)(20,26,23,37)(27,35)(28,34,33,36);(2,5)(6,8)(9,10,12,15,14,17,22,29)(11,16,23,28,21,32,20,33)(13,25,24,39,27,18,35,30)(19,36,26,40,38,34,37,31);(9,11)(10,16)(12,23)(13,26)(14,21)(15,28)(17,32)(18,31)(19,35)(20,22)(24,38)(25,40)(27,37)(29,33)(30,36)(34,39);(1,3,4,7)(2,6,5,8)(10,17)(15,29)(16,32)(18,25)(28,33)(30,39)(31,40)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=ehi,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72313244, #72313273, #72313289, … |
| maximal normal subgroup of | #72313244, #72313273, #72313289, … |
| maximal quotient of | #72343314, #72343360, #72343386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402186);